The official feature list for Mountain Lion says that individual notes opened into their own windows from the Notes app (1.0) remain "pinned" open "even if you close Notes", but this isn't true. Closing Notes closes any notes that have been opened into their own windows.

How do I get the Notes app's note "pinning" feature to work as described.

Double click the name of a note on the overview pane where it lists your notes. It will pop out into its own window. You can then close the main notes window, but the extra window will stay open and act like a stickie note.

To contribute to this discussion, and to the confusion caused by this "feature": yes, indeed, you can close the main notes window, after "pinning" one or more separate notes, but you cannot open it anymore. When you click the notes app, only the then open pinned notes will become active. You have to close all your pinned notes first, if you want to re-open the main notes window in order to create a new note, or to open another one.
This is very annoying indeed.

In addition, when you pin a note, close the main window, then close the pinned notes in order to re-open the main notes window, you may will find your previously pinned notes in double, or even triplicate when you select to view "All Notes". As it turns out, the previously "pinned" notes are now appearing both in their original folder (in my case in iCloud), and in the "recovered items" folder under "On My Mac".